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surfaces and t considered as a function of (x, y, z) will be the characteristic function for them.

The transformation

$X={\frac {x}{r^{2}-c^{2}t^{2}}},\ Y={\frac {y}{r^{2}-c^{2}t^{2}}},\ Z={\frac {z}{r^{2}-c^{2}t^{2}}},\ T={\frac {t}{r^{2}-c^{2}t^{2}}}$

is of special importance because it makes the standard wave surface t = 0 in the original system correspond to a standard wave surface t = 0 in the new system; also, since the equations of the surfaces are

$F(x,\ y,\ z,\ 0)=0,$

and

$F({\frac {x}{r^{2}}},\ {\frac {y}{r^{2}}},\ {\frac {z}{r^{2}}},\ 0)=0,$

respectively, it is clear that one is the inverse of the other with regard to a unit sphere whose centre is at the origin. Our theorem tells us that if the surfaces parallel to the first are given by

$F(x,\ y,\ z,\ t)=0,$

the surfaces parallel to the second are given by

$F\left({\frac {x}{r^{2}-c^{2}t^{2}}},\ {\frac {y}{r^{2}-c^{2}t^{2}}},\ {\frac {z}{r^{2}-c^{2}t^{2}}},\ {\frac {t}{r^{2}-c^{2}t^{2}}}\right)=0.$

Applying this result to the family of right circular cones parallel to a given one, we may obtain the family of Dupin's cyclides parallel to a given one.

To obtain a geometrical interpretation of the transformation we describe a sphere of radius ct round the point (x, y, z) as centre. The inverse sphere is then of radius cT and its centre is at the point (XYZ).

A more general result is that the sphere

$\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}+\left(z-z_{0}\right)^{2}=c^{2}\left(t-t_{0}\right)^{2}$

corresponds in the transformation to the sphere

$\left(X-X_{0}\right)^{2}+\left(Y-Y_{0}\right)^{2}+\left(Z-Z_{0}\right)^{2}=c^{2}\left(T-T_{0}\right)^{2}$,

the centres of the two spheres being corresponding points at the times $t_{0},\ T_{0}$ respectively.

To show that the laws of reflection and refraction remain unchanged in the transformation, we take the surface at which the light is incident as the standard one from which the time is measured. Let $\left(x_{0},\ y_{0},\ z_{0}\right)$ be a point on this surface; then we must associate with this point the time $t_{0}=0$.